One of the scenarios is that the cloud provider has an algorithm they want to keep secret. Clients want to use the algorithm but dont trust cloud provider with their data.

Honestly though, i mostly find FHE interesting theoretically. The fact that its possible at all is black magic. I'm sure if it ever gets remotely efficient people will come up with more creative applications, but in the meanwhile its resesrch worth it for research sake.

> One of the scenarios is that the cloud provider has an algorithm they want to keep secret. Clients want to use the algorithm but dont trust cloud provider with their data.

That would indeed go a long way towards resolving the issues caused by the way SaaS is done. It would let us decouple compute provider from the service, in a way in which it would be the end users who are free to chose where the computation happens, and they'd own all the data by default, while the business secrets and IP of the service provider would still be protected.

Shame to see it's so computationally expensive, I was really hoping this kind of computing would take off.

You nay also want to limit the client’s access to the data: e.g. limit the number or types of queries, which, depending on the encryption type, may require data to be processed on the server.

Your point alludes to the requirements one must consider before implementing private search: query privacy, and bandwidth efficiency. If I need only privacy, then do as you suggest. If I need only bandwidth efficiency, then reveal the query to the server. Thanks for bringing this up.